NUMERACY
Y6
National Numeracy Tests
Reasoning
sample
materials
Reasoning sample materials: Guidance for teachers
The reasoning tests will be first introduced in schools in 2014. It is therefore important that
teachers and learners become increasingly familiar with the requirements in the framework
to identify processes and connections, to represent and communicate, and to review.
Sample items have been produced for each year group to illustrate different question types
and formats for response. Each year group contains one stimulus item, presented through
PowerPoint, which requires information to be shown by the teacher immediately before the
test begins.
The purpose of the stimulus material is to allow learners to engage with unfamiliar contexts.
A teacher script is provided but teachers may use their own words provided no help is given
with the numeracy that is to be assessed.
The sample items are representative of the anticipated level of demand. However, they are not
complete papers: the number of marks within the live tests will be about 20 for each year group,
with one stimulus item followed by between four and eight additional questions. In 2014 each
reasoning test will last 30 minutes. The time taken to deliver the stimulus is in addition to this
assessment time.
How to use the sample items
The sample items can be printed and used for practice before the tests. Strengths and areas
for improvement can then be identified and used to provide additional classroom learning and
teaching activities, where appropriate.
●●
The reasoning sample items can also be used as a basis for classroom discussion, to illustrate
good test techniques. These include the importance of reading the question carefully, where
to write the answers, the importance of showing working to enable others to understand the
reasoning applied, good time management and the benefits of checking answers.
As importantly, the sample items can be used to promote understanding of good responses to
open questions. For example, teachers could anonymise and photocopy a range of responses
and ask learners to work in small groups to rank from ‘best’ to ‘worst’, identifying what is good
about each and why.
Marking of the sample items
A markscheme is provided which is typical of those to be used alongside the live tests.
It includes a range of likely responses with clear guidance on when and how partial credit
should be applied. General marking guidance provides principles of marking to facilitate
consistency across schools.
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2
Presentation to be shown to learners before doing question 1
The text in the right-hand boxes should be read to learners. Teachers can use their own words,
or provide additional explanation of contexts, if necessary. However, no help must be given with
the numeracy that is to be assessed.
Slide 1
Do you know where this photograph was taken?
It’s the Millennium Stadium in Cardiff. Newspaper
reports said about eighty thousand people
watched the match that is shown in the
photograph. How would they know how many
people were there?
(Discuss. Learners are likely to refer to tickets sold,
entry gates, etc. Draw out that this allows the
organisers to be able to know the exact number
even though it has been rounded.)
This photograph shows another crowd, but this
time it was in Monmouth. The photograph shows
people waiting to see the Carnival procession.
Slide 2
There were far too many people to count them
individually, and there were no tickets sold, so we
need to think about how to estimate the number
of people in the crowd.
Slide 3
77 000?
150 000?
Whenever you see an estimate of the number of
people in a crowd you need to think about who
has made the estimate.
For example, there was a rally in Hong Kong
remembering Chinese people who died. Here
are two different estimates of the number of
people who attended. One estimate was made by
the people who organised the event. The other
estimate was made by the police. Which do you
think was made by the organisers of the event?
Discuss, drawing out that organisers may
over-estimate to show how much support they have.
3
Slide 4
Estimating the number of people in a crowd is
difficult and mathematicians and scientists have
spent a long time trying to find the best method.
1m
1
1m
1m
2
One way is to think about how much space
people take up.
1m
For light crowds – that means crowds where
people are not too jammed together – they
estimate that each person takes up about this
amount of space, a square that is one metre by
one metre (point to the left hand square).
But when the crowds are more tightly packed
together, they estimate that two people will fit
in the same space. So for each square that is one
metre by one metre, they estimate there will be
two people.
Slide 5
You are going to use that information to estimate
the size of a crowd that stood along a wide road
in London called The Mall. They were standing
there in front of Buckingham Palace for the royal
wedding of Prince William and Kate Middleton.
All the information you need is in your booklet.
When you have finished there are other questions
to answer.
Remember that for some of the questions you
will need to use your calculator, and it is very
important to show your working so that someone
else can understand what you are doing and why.
4
1
Estimate the number of people standing on The Mall for the
Royal Wedding.
Assume:
●●
two people for each square, 1 metre by 1 metre
●●
the Mall is 30 metres wide.
Use the map on the opposite page and explain each step of your
reasoning.
Give your answer to the nearest thousand people.
4m
5
Buckingham Palace
TH
E
M
AL
L
100m
6
2
Ambulances have
not
on the front.
Why?
1m
3
This quilt cover is made from
two red and two yellow squares.
What fraction of the quilt cover is red?
2m
7
4
Plan of the
inside of a plane
Each green rectangle
shows a seat.
How many seats are there?
1m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
What is the quickest way to work out
this number?
1m
It costs a lot of money to fly a plane.
They must get at least £25 000 for tickets.
So far, 80 people have bought tickets.
Each ticket costs £250
How many more tickets must they sell
to get £25 000?
more tickets
3m
8
Reasoning sample materials: Marking guidance
It is important that the tests are marked accurately. The questions and answers below help
to develop a common understanding of how to mark fairly and consistently.
Must learners use the answer boxes?
Provided there is no ambiguity, learners can respond anywhere on the page. If there is more
than one answer the one in the answer box must be marked, even if incorrect. However, if
the incorrect answer is clearly because of a transcription error (e.g. 65 has been copied as 56),
mark the answer shown in the working.
●●
What if learners use a method that is not shown within the markscheme?
The markschemes show the most common methods, but alternative approaches may deserve
credit − use your professional judgement. Any correct method, however idiosyncratic,
is acceptable.
●●
Does it matter if the learner writes the answer differently from that shown in the
markscheme?
Numerically equivalent answers (e.g. eight for 8, or two quarters or 0.5 for half) should be
marked as correct unless the markscheme states otherwise.
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How should I mark answers involving money?
Money can be shown in pounds or pence, but a missing zero, e.g. £4.7, should be marked
as incorrect.
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How should I mark answers involving time?
In the real world, specific times are shown in a multiplicity of ways so accept, for example,
02:30, 2.30, half past 2, etc. Do not accept 2.3 as this is ambiguous. The same principle should
be used for marking time intervals, e.g. for two and a half hours accept 2.5 but not 2.5pm.
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What if the method is wrong but the answer is correct?
Unless the markscheme states otherwise, correct responses should be marked as correct even
if the working is incorrect as learners may have started again without showing their revised
approach.
●●
What if the learner has shown understanding but has misread information in the question?
For a two (or more) mark item, if an incorrect answer arises from misreading information given
in the question and the question has not become easier as a result then deduct one mark only.
For example, if the 2 mark question is 86 × 67 and the learner records 96 × 67 then gives the
answer 6432, one mark only should be given. In a one mark question, no marks can be given.
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What should I do about crossed out work?
Working which has been crossed out and not replaced can be marked if it is still legible.
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What is the difference between a numerical error and a conceptual error?
A numerical error is one in which a slip is made, e.g. within 86 × 67 the learner works out
6 × 7 = 54 within an otherwise correct response. A conceptual error is a more serious
misunderstanding for which no method marks are available, for example if 86 × 60
is recorded as 516 rather than 5160
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9
Year 6 Reasoning sample materials: Markscheme
Q
Marks
1
4m
Answer
Comments
Shows the following steps to justify their
answer of 56 000 (or 55 000, see comments
section)
Throughout, condone units
that are incorrect or omitted
1.Measures and uses the scale to convert
to metres
Accept 920 to 940m inclusive
2.Multiplies by 30 to find the area of the
Mall
Accept 27 600 to 28 200
inclusive
3.Doubles to find the number of people
Accept 55 200 to 56 400
inclusive
Must follow from their number
of people, i.e. from 55 200 their
answer must be 55 000
4.Rounds to the nearest thousand
e.g.
●●
Or 3m
18.6cm is 930m
930 × 30 = 27 900m2
2 people per m2 so 55 800
Answer 56 000 people
Shows a value between 54 000 and 57 000
inclusive
Or
Rounds their (double their incorrect
length × 30) to the nearest ten thousand
Or 2m
Shows a value between 27 600 and 28 200
inclusive
Or
Doubles their (incorrect length × 30)
Or 1m
Shows a value between 920 and 940
inclusive
Or
Multiplies their incorrect length in metres
by 30
10
Q
Marks
2
1m
Answer
Comments
Gives an explanation that includes drivers
and a mirror, e.g.
Do not accept partial
explanations, e.g.
●●
●●
3
2m
Or 1m
So drivers can see the word ambulance in
their mirror
So that when you are in your car you see
it the right way when the ambulance is
behind (mirror implied)
●●
So drivers can see it
●●
Because of the mirror
One sixth, or equivalent fraction
Shows that the large yellow square contains
nine of the smaller yellow squares, e.g.
●●
11
Lines need not be accurate
provided the learner’s intention
is clear
Q
Marks
4i
1m
148
4ii
1m
Groups seats, using multiplication to give a
correct calculation, e.g.
4iii
3m
Or 2m
Answer
Comments
●●
25 × 6 − 2 [or 50 × 3 − 2]
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24 × 6 + 4 [or 48 × 3 + 2 + 2]
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11 × 6 + 4 + 13 × 6
●●
11 × 3 + 11 × 3 + 4 + 13 × 3 + 13 × 3
●●
20
Shows that another £5000 of tickets must
be sold
Or
Shows a complete correct method even if
there are numerical errors, e.g.
●●
Or 1m
80 × 250 = 18 000 (error)
25 000 − 18 000 = 7 000
7 000 ÷ 250
Shows that the total sales to date is £20 000
Or
Shows a correct method to find the total
amount left to be sold, e.g.
●●
Provided a correct method
is shown, accept numerical
errors, e.g.
80 × 250 = 18 000 (error)
25 000 − 18 000 = 7 000
Or
Shows their total amount left to be
sold ÷ 250
12
24 × 6 = 124 (error),
124 + 4 = 128
Do not accept repeated
addition
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